Bivariate Normal

Correlation 0

Correlation -0.5

Correlation 0.9

Probabilities by Integration

\[P(0.5 \leq X_1 \leq 1.5 \text{ and } 0 \leq X_2 \leq 1) = \int_{0.5}^{1.5} \int_{0}^1 f(x_1, x_2 | \mathbf{\mu}, \Sigma) \, d x_2 d x_1\]

Trivariate Normal

In three dimensions, we can’t plot the density function directly - but we can visualize the distribution via its equi-density surfaces, which are analogous to the ellipses for the bivariate normal distribution.

Here’s a plot of equi-density surfaces of the \(\text{Normal}\left(\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 & 0.3 & 0.6 \\ 0.3 & 2 & 1.1 \\ 0.6 & 1.1 & 3\end{bmatrix}\right)\) distribution.